Immersed Legendrian knots satisfy an h-principle  [1]}, and the rotation class classifies them up to regular Legendrian homotopy. If we have two Legendrian knots which are smoothly homotopic, we can compare their rotation classes as follows. A formal Legendrian defines an isomorphism \(\xi |_L \cong TL \otimes \mathbb {C}\) , therefore two formal Legendrians together define a difference element in \(\operatorname{Aut}_{\mathbb {C}}(TL \otimes \mathbb {C})\) , also known as the gauge group of \(\xi |_L\) . Two Legendrians have the same rotation class if and only if this difference element is in the component of the identity. If \(\xi |_L\)  is trivial (which is always the case if \(\xi |_Y\)  is trivial) then \(\operatorname{Aut}_{\mathbb {C}}(\xi |_L) \cong \operatorname{Map}(L, U_n)\) , thus the difference class \(r(L_0) - r(L_1)\)  is an element of \(K^1(L)\)  in this case.
